Journal of Applied Mathematics
Volume 2020 (2020), Article ID 7290403, 5 pages
https://doi.org/10.1155/2020/7290403

Research Article

Explicit Expression for Arbitrary Positive Powers of Special Tridiagonal Matrices

Mohammad Beiranvand and Mojtaba Ghasemi Kamalvand

Department of Mathematics, Lorestan University, Khorramabad, Iran

Correspondence should be addressed to Mojtaba Ghasemi Kamalvand; ghasemi.m@lu.ac.ir

Received 18 May 2020; Revised 23 July 2020; Accepted 27 July 2020; Published 12 September 2020

Academic Editor: Igor Andrianov

Copyright © 2020 Mohammad Beiranvand and Mojtaba Ghasemi Kamalvand. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Tridiagonal matrices appear frequently in mathematical models. In this paper, we derive the positive integer powers of special tridiagonal matrices of arbitrary order.

1. Introduction

Recently, computing the integer powers of tridiagonal matrices has been a very popular problem. Tridiagonal matrices are used in different areas of science and engineering, for instance, the solution of difference systems [1], numerical solution of PDEs [2], telecommunication system analysis [3, 4], texture modeling [5], and image processing and coding [6]. In these areas, the computation of the powers of these matrices is necessary. Therefore, there are a lot of studies dealing with the powers of these matrices using the well-known expression where is Jordan’s form of the matrix and is the transforming matrix. We need the eigenvalues and eigenvectors of these matrices to calculate .

Rimas [7–13], investigated positive integer powers of certain tridiagonal matrices, Oteles and Akbulak [14, 15] and Gutierrez [16] generalized some papers of Rimas, and Wang [17] derived the entries of positive integer powers of complex persymmetric anti-tridiagonal matrices with constant anti-diagonals. Some authors also investigated integer powers of certain tridiagonal matrices.

In this paper, we consider the -th order tridiagonal matrix of the following type

where , , , , and are the numbers in the complex . There are many mathematical models that are involved in this form [18].

2. Main Results

According to the following lemmas, Wen-Chyuan Yueh obtains eigenvalue and corresponding eigenvectors for matrix (1), in special cases. In this section, we firstly find the transforming matrices and their invers for the matrix (1). Secondly, we present a general expression for the entries of for .

Lemma 1 (see [19]). Suppose , , and , then the eigenvalues of are given by

the corresponding eigenvectors , are given by

In this paper, we need the following theorem.

Theorem 2. (canonical Jordan’s form [20]). Let be any square matrix. Then, there exists a nonsingular matrix which transforms into a block diagonal matrix such that

which is called the canonical Jordan’s form, being the eigenvalues of and a Jordan block of the form

Since all the eigenvalues for are distinct ( is a strictly decreasing function of on (0,), and ), columns of the transforming matrix are the eigenvectors of the matrix (1). Also, all eigenvalues correspond to the single Jordan cell in the matrix , then we write down Jordan’s form of the matrix as

From (3), we can write the column transforming matrix as

for .

Hence,

Let

in which

for .

Therefore, we have

So an explicit expression for will suffice.

Theorem 3. Suppose is defined as above, then

Proof. We show that

From (10), equals and

follows

If ,

If ,

If and are even or odd, then and are even; therefore, , so we have

If one of or is even and the other is odd, then and are odd; therefore, we have

From (18), (20), and (21) follows

For derivation of the formula for the entries of from (4), (6), (10), (12), and (13), we can conclude

By substituting and in the latter equation and doing the necessary computation follows

for , where

Lemma 4 (see [19]). Suppose and and , then the eigenvalues of are given by

The corresponding eigenvectors, , are given by

From (26) for we can write the columns transforming matrix as

Hence,

Let

in which

for

Therefore, we have

So an explicit expression for will suffice.

Theorem 5. Suppose is defined as above, then

Proof. We show that

From (30) equals and

follows

If ,

If ,

If and are even or odd, then is even and is odd; therefore,

So we have

If one of or is even and the other is odd, then and are even; therefore, we have

and

Therefore,

Similar for (24), we can conclude

for .

Lemma 6 (see [18]). Suppose and , then the eigenvalues of are given by

The corresponding eigenvectors are given by

In the case , the eigenvalues are given by (44) and the corresponding eigenvectors by

In these cases which are similar to previous cases, we showed that integer powers of the matrix for respectively,

where , and

where

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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